Is there a signature scheme which doesn't rely on the difficulty of factoring/dlp which generates short signatures?

Question

The Lamport signature scheme, for example, doesn't rely on the difficulty of any problem and it only depends on the existence of one-way functions.
Is there an alternative scheme which also doesn't rely on the difficulty of factoring/discrete logarithm, but provides shorter signatures?

Answer 1

There are actually quite a few of these. Interest has been raised on this topic mainly due to the "post-quantum" security of such schemes. Also, Lamport is only a one-time signature, and we want a full-blown signature schemes.

For just one example see, SPHINCS: https://sphincs.cr.yp.to/sphincs-20150202.pdf.

Answer 2

As @YehudaLindell says, since Shor's Quantum Factoring Algorithm kills factoring and discrete log based cryptosystems, almost everything in the "Post-Quantum" crypto world will meet your requirement.

SPHINCS is similar to the Lamport scheme in that it only relies on hash functions.

There's also the realm of lattice-based cryptosystems such as NTRU. Also systems based on isogenies on supersingular elliptic curves produce small signatures.

Links to overview reports on various post-quantum crypto primitives:

• NIST DRAFT Report on Post-Quantum Cryptography
• NCC Group Cryptographic Services - Post-quantum cryptography overview